Now that we know how Riemann Sums are a way for us to evaluate the area under a curve, which is to divide the region into rectangles of fixed width and adding up the areas, let’s look at the Definition of a Definite Integral as it pertains to Sigma Notation and the Limit of Finite Sums.
In other words, when we say Sigma Notation and Limits of Finite Sums, it is nothing more than the formal definition of a Riemann Sum and the Definite Integral
What we are about to do is to take a function and express it as the limit of a sequence of Riemann Sums over an interval.
What’s different about these types of questions than the ones that we saw in the previous video dealing with Riemann Sums is that we are not given a specified number of subintervals. This means we will need to create an interval, as Khan Academy nicely states, that will become infinitely small.
But don’t worry, the process is straightforward with only three steps:
- Find your width (change in x) for n subintervals
- Find your right endpoint
- Plug everything into your function and evaluate using summation formulas and your algebra skills
Rules for Working with Sigma Notation
Together we will look at two questions where we are going to use the Definition of the Definite Integral to find an expression for the area under a curve as a limit.
Next, we will look at three questions where we will evaluate our expression by utilizing our Sigma Notation and Rules for Finite Sums.
Sigma Notation – Video
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